Narrow Results

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal $z$-degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology.

We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of $L$ is given by $P_L(-v^{-1},z)$. This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial.

This paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram $D$, it is known that $s(D)$, the number of Seifert circles in $D$, equals the braid index $b(D)$ of $D$ if $D$ contains no lone crossings, where a crossing in $D$ is called a lone crossing if it is the only crossing between two Seifert circles in $D$. If $D$ contains lone crossings, then one can reduce the number of Seifert circles in $D$ using link-type preserving moves such as the Murasugi–Przytycki operation. Let $r(D)\ge 1$ be the maximum number of Seifert circles in $D$ that can be reduced, then we have $b(D)\le s(D)-r(D)$. On the other hand, if the $a$-span of the HOMFLY polynomial of $D$ satisfies the equality $a$-span$/2+1=s(D)-r(D)$, then the Morton–Frank–Williams (MFW) inequality $a$-span$/2+1\le b(D)$ leads us to the simple braid index formula $b(D)=s(D)-r(D)$. In this paper, we derive explicit formulas for many alternating links based on minimum projections of these links by establishing the equality $a$-span$/2+1=s(D)-r(D)$. Our methods depend on the structures of the link diagrams under consideration and our results lead to explicit braid index formulas that are applicable to a very large class of links, a proper subset of which contains all two bridge links, all alternating pretzel links, and more generally all alternating Montesinos links. The derived braid index formula for an alternating Montesinos link is a function whose inputs are the signs of the crossings in the rational tangles of the Montesinos link. Finally, by applying the now proven Jones Conjecture on the writhe of minimum braids of a link, our results also allow us to obtain explicit formulas of the writhe of minimum braids for the links discussed in this paper from the minimum projections of these links.

This paper is the second installment of our in-depth study on the braid index problem of pretzel links. We classify pretzel links into three categories based on the Seifert circle decompositions in their standard link diagrams. In our initial paper, we successfully identified the braid indices for all Type 1 and Type 2 pretzel links. In this paper, we share our findings on Type 3 pretzel links. We are able to determine the exact braid indices for most Type 3 pretzel links. For the small subset of Type 3 pretzel links where exact braid indices remain undetermined, we establish their braid indices within a range of two consecutive integers. With some numerical evidence, we conjecture that in such an exceptional case, the braid index of the Type 3 pretzel link is given by the larger of the two consecutive integers given by our formulas.

In this paper and its sequel, we study the braid indices for all non-alternating pretzel links by a systematic approach. Pretzel links can be divided into three types according to the Seifert circle decompositions of their standard link diagrams. In this paper, we present our results on Type 1 and Type 2 pretzel links, namely the complete formulation of the braid indices for all non-alternating Type 1 and Type 2 pretzel links. Since the braid indices are already known for all alternating pretzel links from our previous work, it means that we have now completely determined the braid indices for all Type 1 and Type 2 pretzel links.

We study the relation between the two notions of index of graphs defined in [1]. We show that in the geometrically significant context of bipartite graphs the two coincide.